The Bulletin of Symbolic Logic

We discuss the typography of the notation used by Gottlob Frege in his Grundgesetze der Arithmetik. §1. Background to the Grundgesetze der Arithmetik. Grundgesetze der Arithmetik was to have been the pinnacle of Gottlob Frege’s life’s work — a rigorous demonstration of how the fundamental laws of classical pure mathematics of the natural and real numbers can be derived from principles which, in Frege’s view,were purely logical.His logical system, calledBegriffsschrift, i.e., “concept-script”, was first introduced in 1879 in his book with this title [19]. It includes the first occurrence in formal logic of quantifiers,1 with which multiple and embedded generality could be expressed — no earlier logical system was capable of this. It also offers the first formulation of a logical system that contains relations rather thanmerelymonadic predicates. In addition, Frege here presents his celebrated definition of the ancestral of a relation. Taken together, these developments made logic expressively adequate for mathematics for the first time in history [13, pp. xxxv–xxxvi]. Begriffsschrift is thus widely acknowledged as the greatest advance in logic since Aristotle — as W.V. Quine put it [43, p. vii]: Logic is an old subject, and since 1879 it has been a great one. In 1884 Frege published the book Die Grundlagen der Arithmetik [20] in which he formulates and argues for logicism— the idea that arithmetic (and analysis) is reducible to logic. The principal aim of Grundlagen is to provide philosophical arguments for logicism, but Frege also offers proof-sketches of how Peano’s axioms for arithmetic can be derived from entirely logical principles, based on an explicit definition of “cardinal number” and taking extensions of concepts as primitive. It was to be the task of hismagnumopus, Grundgesetze der Arithmetik [21,23], to show conclusively the purely logical nature of mathematics by presenting gapless proofs of the axioms of arithmetic and real analysis in his formal system, using only explicit definitions Received October 21, 2014. 2010 Mathematics Subject Classification. 03A05, 01A55, 03Bxx, 00A30.